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Shaoning Han

Publications and source records attributed to Shaoning Han.

14 recordsLinked to original sources
arXiv2026

Convexification of mixed-integer quadratic optimization via decision diagrams

We study mixed-integer quadratic optimization (MIQO) problems with indicator variables. We propose a unified framework, based on decision diagrams, that serves both to solve the associated optimization problems and to construct ideal conic quadratic extended formulations of the closure of the convex hull of the underlying mixed-integer set. The construction applies to arbitrary quadratics and to any combinatorial constraints admitting a tractable dynamic programming representation. The resulting diagrams and the ensuing convex hull descriptions are of polynomial size when the quadratic is low-rank, or when the support graph of the Hessian or of its inverse is a tree, recovering and generalizing several results from the literature. For structured sparse and inverse-sparse quadratics, we show that approximate decision diagrams have size linear in the dimension while yielding solutions with arbitrarily low optimality gap. Computational experiments demonstrate the effectiveness of the proposed approach.

math.OC↗
arXiv2026

A Geometric Perspective on Polynomially Solvable Convex Maximization

Convex maximization encompasses a broad class of optimization problems and is generally NP-hard, even for low-rank objectives. This paper investigates structural conditions under which convex maximization becomes polynomially solvable. From a geometric perspective, we introduce comonotonicity, a structural property of the feasible region crucial for problem tractability, and establish mathematical characterizations of this property. Under comonotonicity and mild additional assumptions, we develop a unified enumerative framework showing that fixed-rank convex maximization is polynomially solvable. This viewpoint recovers several known tractability results that previously required separate analyses, such as fixed-rank convex matroid maximization and sparse principal component analysis (SPCA). Furthermore, for the more structured class of standard comonotone feasible regions, we refine the analysis via a lifting technique to achieve a square-root improvement in the complexity bound. Finally, applications to SPCA and its variants illustrate the broad applicability and effectiveness of the proposed framework.

math.OC↗
arXiv2025

Convex Submodular Minimization with Indicator Variables

We study a general class of convex submodular optimization problems with indicator variables. Many applications such as the problem of inferring Markov random fields (MRFs) with a sparsity or robustness prior can be naturally modeled in this form. We show that these problems can be reduced to binary submodular minimization problems, possibly after a suitable reformulation, and thus are strongly polynomially solvable. %We also discuss the implication of our results in the case of quadratic objectives. Furthermore, we develop a parametric approach for computing the associated extreme bases under certain smoothness conditions. This leads to a fast solution method, whose efficiency is demonstrated through numerical experiments.

math.OC↗
arXiv2025

Rank-one convexification for quadratic optimization problems with step function penalties

We investigate convexification for convex quadratic optimization with step function penalties. Such problems can be cast as mixed-integer quadratic optimization problems, where binary variables are used to encode the non-convex step function. First, we derive the convex hull for the epigraph of a quadratic function defined by a rank-one matrix and step function penalties. Using this rank-one convexification, we develop copositive and semi-definite relaxations for general convex quadratic functions. Leveraging these findings, we construct convex formulations to the support vector machine problem with 0--1 loss and show that they yield robust estimators in settings with anomalies and outliers.

math.OC↗
arXiv2024

Improving the Solution of Indefinite Quadratic Programs and Linear Programs with Complementarity Constraints by a Progressive MIP Method

Indefinite quadratic programs (QPs) are known to be very difficult to be solved to global optimality, so are linear programs with linear complementarity constraints. Treating the former as a subclass of the latter, this paper presents a progressive mixed integer linear programming method for solving a general linear program with linear complementarity constraints (LPCC). Instead of solving the LPCC with a full set of integer variables expressing the complementarity conditions, the presented method solves a finite number of mixed integer subprograms by starting with a small fraction of integer variables and progressively increasing this fraction. After describing the PIP (for progressive integer programming) method and its various implementations, we demonstrate, via an extensive set of computational experiments, the superior performance of the progressive approach over the direct solution of the full-integer formulation of the LPCCs. It is also shown that the solution obtained at the termination of the PIP method is a local minimizer of the LPCC, a property that cannot be claimed by any known non-enumerative method for solving this nonconvex program. In all the experiments, the PIP method is initiated at a feasible solution of the LPCC obtained from a nonlinear programming solver, and with high likelihood, can successfully improve it. Thus, the PIP method can improve a stationary solution of an indefinite QP, something that is not likely to be achievable by a nonlinear programming method. Finally, some analysis is presented that provides a better understanding of the roles of the LPCC suboptimal solutions in the local optimality of the indefinite QP.

math.OC↗
arXiv2024

Real-time solution of quadratic optimization problems with banded matrices and indicator variables

We consider mixed-integer quadratic optimization problems with banded matrices and indicator variables. These problems arise pervasively in statistical inference problems with time-series data, where the banded matrix captures the temporal relationship of the underlying process. In particular, the problem studied arises in monitoring problems, where the decision-maker wants to detect changes or anomalies. We propose to solve these problems using decision diagrams. In particular we show how to exploit the temporal dependencies to construct diagrams with size polynomial in the number of decision variables. We also describe how to construct the convex hull of the set under study from the decision diagrams, and how to deploy the method online to solve the problems in milliseconds via a shortest path algorithm.

math.OC↗
arXiv2024

Robust support vector machines via conic optimization

We consider the problem of learning support vector machines robust to uncertainty. It has been established in the literature that typical loss functions, including the hinge loss, are sensible to data perturbations and outliers, thus performing poorly in the setting considered. In contrast, using the 0-1 loss or a suitable non-convex approximation results in robust estimators, at the expense of large computational costs. In this paper we use mixed-integer optimization techniques to derive a new loss function that better approximates the 0-1 loss compared with existing alternatives, while preserving the convexity of the learning problem. In our computational results, we show that the proposed estimator is competitive with the standard SVMs with the hinge loss in outlier-free regimes and better in the presence of outliers.

cs.LG↗
arXiv2022

Convex Submodular Minimization with Indicator Variables

We study a general class of convex submodular optimization problems with indicator variables. Many applications such as the problem of inferring Markov random fields (MRFs) with a sparsity or robustness prior can be naturally modeled in this form. We show that these problems can be reduced to binary submodular minimization problems, possibly after a suitable reformulation, and thus are strongly polynomially solvable. Furthermore, we develop a parametric approach for computing the associated extreme bases under certain smoothness conditions. This leads to a fast solution method, whose efficiency is demonstrated through numerical experiments.

math.OC↗
arXiv2021

Some Strongly Polynomially Solvable Convex Quadratic Programs with Bounded Variables

This paper begins with a class of convex quadratic programs (QPs) with bounded variables solvable by the parametric principal pivoting algorithm with $\mathcal{O}(n^3)$ strongly polynomial complexity, where $n$ is the number of variables of the problem. Extension of the Hessian class is also discussed. Our research is motivated by a recent reference [7] wherein the efficient solution of a quadratic program with a tridiagonal Hessian matrix in the quadratic objective is needed for the construction of a polynomial-time algorithm for solving an associated sparse variable selection problem. With the tridiagonal structure, the complexity of the QP algorithm reduces to $\mathcal{O}(n^2)$. Our strongly polynomiality results extend previous works of some strongly polynomially solvable linear complementarity problems with a P-matrix [9]; special cases of the extended results include weakly quasi-diagonally dominant problems in addition to the tridiagonal ones.

math.OC↗
arXiv2021

Compact extended formulations for low-rank functions with indicator variables

We study the mixed-integer epigraph of a special class of convex functions with non-convex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions Extended formulations describing the convex hull of such sets can easily be constructed via disjunctive programming, although a direct application of this method often yields prohibitively large formulations, whose size is exponential in the number of variables. In this paper, we propose a new disjunctive representation of the sets under study, which leads to compact formulations with size exponential in the dimension of the nonlinear function, but polynomial in the number of variables. Moreover, we show how to project out the additional variables for the case of dimension one, recovering or generalizing known results for the convex hulls of such sets (in the original space of variables). Our computational results indicate that the proposed approach can significantly improve the performance of solvers in structured problems.

math.OC↗
arXiv2020

Fractional 0-1 programming and submodularity

In this note we study multiple-ratio fractional 0--1 programs, a broad class of NP-hard combinatorial optimization problems. In particular, under some relatively mild assumptions we provide a complete characterization of the conditions, which ensure that a single-ratio function is submodular. Then we illustrate our theoretical results with the assortment optimization and facility location problems, and discuss practical situations that guarantee submodularity in the considered application settings. In such cases, near-optimal solutions for multiple-ratio fractional 0--1 programs can be found via simple greedy algorithms.

math.OC↗
arXiv2020

2x2 convexifications for convex quadratic optimization with indicator variables

In this paper, we study the convex quadratic optimization problem with indicator variables. For the bivariate case, we describe the convex hull of the epigraph in the original space of variables, and also give a conic quadratic extended formulation. Then, using the convex hull description for the bivariate case as a building block, we derive an extended SDP relaxation for the general case. This new formulation is stronger than other SDP relaxations proposed in the literature for the problem, including Shor's SDP relaxation, the optimal perspective relaxation as well as the optimal rank-one relaxation. Computational experiments indicate that the proposed formulations are quite effective in reducing the integrality gap of the optimization problems.

math.OC↗
arXiv2018

Sparse and Smooth Signal Estimation: Convexification of L0 Formulations

Signal estimation problems with smoothness and sparsity priors can be naturally modeled as quadratic optimization with $\ell_0$-"norm" constraints. Since such problems are non-convex and hard-to-solve, the standard approach is, instead, to tackle their convex surrogates based on $\ell_1$-norm relaxations. In this paper, we propose a new iterative (convex) conic quadratic relaxations that exploit not only the $\ell_0$-"norm" terms, but also the fitness and smoothness functions. The iterative convexification approach substantially closes the gap between the $\ell_0$-"norm" and its $\ell_1$ surrogate. These stronger relaxations lead to significantly better estimators than $\ell_1$-norm approaches and also allow one to utilize affine sparsity priors. In addition, the parameters of the model and the resulting estimators are easily interpretable. Experiments with a tailored Lagrangian decomposition method indicate that the proposed iterative convex relaxations \rev{yield solutions within 1\% of the exact $\ell_0$ approach, and can tackle instances with up to 100,000 variables under one minute.

stat.ML↗